Rigidity of Solvable Group Actions
This thesis investigates dynamical properties of actions of abelian-by-cyclic groups on compact manifolds. For a non-singular integer matrix A, let FA be the fundamental group of the mapping cylinder of the induced map fA on the torus T n. The standard actions p), of FA on the circle RP 1 are generated by maps f(x) = Ax and gi(x) = x + bi, where A is a real-valued eigenvalue for A, and (01, ..., 0n) is the associated eigenvector. It is known that any analytic action of FA on the circle is a ramified lift of one of the standard actions p),. This thesis shows that for each analytic action, p, there exists R > 2 such that p is Cr locally rigid for all r > R. We then consider actions of the groups FA on compact manifolds of higher dimension that are generated by C1 diffeomorphisms close to the identity. We show that any action of FA on a surface with non-zero Euler characteristic has a global fixed point. Also, we show that for any compact manifold M, there are no faithful actions of the Baumslag-Solitar group generated by diffeomorphisms close to the identity.